Integrated Nested Laplace Approximations (INLA) Demonstration

Kenneth A. Flagg

Motivation

Poisson Processes and Sampling

  • Modeled as (log of) latent Gaussian Process

Poisson Processes and Sampling

  • Observe points
  • Make inferences about intensity function

Poisson Processes and Sampling

  • Not widely done; some work on point process models for distance sampling data (Johnson, Laake, and VerHoef 2014, Yuan et al. (2017))

INLA Introduction

Why INLA?

  • H. Rue, Martino, and Chopin (2009)
  • Bayesian Hierarchical models
    • Many latent Gaussian variables
    • Few parameters
    • E.g. spatial prediction using Gaussian process model
  • Monte Carlo methods impractical

Why INLA?

  • Approximate posterior marginals using Laplace expansions
    • Clever algebra, end up with posterior in denominator
    • Taylor expand log-posterior about its mode
    • Results in a Gaussian approximation
    • Some numerical integration needed

Advantages and Disadvantages

  • Advantages
    • Accurate approximation
    • Fast computation of many posterior marginals
  • Disadvantages
    • Does not provide full joint posterior
    • Slow with >4-6 parameters

Normal Example

Normal Example

Blangiardo and Cameletti (2015) section 4.9

  • \(\mathbf{y} = (y_{1}, \dots, y_{n})'\) independent Gaussian observations
  • \(y_{i} \sim \mathsf{N}(\theta, \sigma^{2})\)
  • \(\theta \sim \mathsf{N}(\mu_{0}, \sigma_{0}^{2})\)
  • \(\psi = 1/\sigma^{2}\), \(\psi \sim \mathrm{Gamma}(a, b)\)

The posterior distribution of the nuisance parameter is

\[p(\psi|\mathbf{y}) \propto \frac{p(\mathbf{y} | \theta, \psi) p(\theta) p(\psi)} {p(\theta | \psi, \mathbf{y})}\]

Normal Example

  • Priors: \(\mu_{0} = -3\), \(\sigma_{0}^{2} = 4\), \(a = 1.6\), \(b = 0.4\)
  • 30 observed points

Normal Example

Normal Example

Normal Example

Normal Example

Spatial Point Pattern: Trees in a Rainforest

Spatial Point Pattern

  • Beilschmiedia pendula Lauraceae locations in a plot in Panama (Møller and Waagepetersen 2007)
  • bei dataset in spatstat (Baddeley and Turner 2005)

Spatial Point Pattern

  • Model from Møller and Waagepetersen (2007)
    • Log-intensity: \(\log(\Lambda(u)) = \beta + \Psi(u)\)
    • \(\Psi\) a GP with exponential covariariance, variance \(\sigma^{2}\), and practical range \(\alpha\)
  • Priors
    • \(\beta \sim \mathrm{Unif}(\infty, \infty)\)
    • \(\sigma \sim \mathrm{Unif}(0.001, \infty)\)
    • \(\log(\alpha) \sim \mathrm{Unif}(1, 235)\)

Spatial Point Pattern

Spatial Point Pattern

Spatial Point Pattern

Spatial Point Pattern

Spatial Point Pattern

Spatial Point Pattern

Things to Address

Things to Address

  • inlabru has a nice interface but is slow and poorly documented
  • Simpson et al. (2016) present a faster method
  • Sampling region is easy to define in R-INLA
  • Evaluate performance
  • Then address where to sample

References

References

Baddeley, Adrian, and Rolf Turner. 2005. “Spatstat: An R Package for Analyzing Spatial Point Patterns.” Journal of Statistical Software 12 (6): 1–42.

Blangiardo, Marta, and Michela Cameletti. 2015. Spatial and Spatio-Temporal Bayesian Models with R-INLA. Wiley.

Johnson, Devin, Jeff Laake, and Jay VerHoef. 2014. DSpat: Spatial Modelling for Distance Sampling Data. https://CRAN.R-project.org/package=DSpat.

Møller, J, and RP Waagepetersen. 2007. “Modern Spatial Point Process Modelling and Inference.” Scandinavian Journal of Statistics 34: 643–711.

Rue, Håvard, Sara Martino, and Nicolas Chopin. 2009. “Approximate Bayesian Inference for Latent Gaussian Models by Using Integrated Nested Laplace Approximations.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 71 (2): 319–92.

Simpson, Daniel, Janine B Illian, Finn Lindgren, Sigrunn H Sørbye, and Havard Rue. 2016. “Going Off Grid: Computationally Efficient Inference for Log-Gaussian Cox Processes.” Biometrika 103 (1): 49–70.

Yuan, Yuan, Fabian E Bachl, Finn Lindgren, David L Borchers, Janine B Illian, Stephen T Buckland, Haavard Rue, Tim Gerrodette, and others. 2017. “Point Process Models for Spatio-Temporal Distance Sampling Data from a Large-Scale Survey of Blue Whales.” The Annals of Applied Statistics 11 (4): 2270–97.